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Is there a name for a property that only needs to be checked for either prime or maximal ideals in order to show that it holds for all ideals? An example would be being a principal ideal for which it's enough to check that all prime ideals are principal.

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I think if there were a single expression it must just simply be "test set."

I came to this conclusion because I see it used both in Noncommutative generalizations of theorems of Cohen and Kaplansky by Reyes, and also in A note on prime ideal which test injectivity by Beachy and Weakley.

If you only want to restrict your attention to the commutative case, then you will want to know about the other related papers that preceded the first linked paper, so let me include those too:

  • Lam, T. Y.; Reyes, Manuel L. A prime ideal principle in commutative algebra. J. Algebra 319 (2008), no. 7, 3006–3027.

  • Lam, T. Y.; Reyes, Manuel L. Oka and Ako ideal families in commutative rings. Rings, modules and representations, 263–288, Contemp. Math., 480, Amer. Math. Soc., Providence, RI, 2009.

I highly recommend checking out the first linked paper above if you are interested in the idea of "test sets." You will want to pay special attention to section 3, which begins this way:

In this section we develop an appropriate notion of a “test set” for certain properties of right ideals in noncommutative rings.

It goes on to give a wonderful setup that really gets at the heart of how test sets work for the classical theorems (and more.)

I haven't read the last bulleted items in a year, but I believe the first one will also talk about what you want to hear in detail for commutative rings.

All of these papers are great, and I simply cannot stop plugging Noncommutative generalizations.... I basically recommend it to anyone who will listen.

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